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C: Physical Processes in Nanomaterials and Nanostructures

Carrier Dynamics and Interactions for Bulk-Like Photoexcitation of Colloidal Indium Arsenide Quantum Dots Austin P. Spencer, William K. Peters, Nathan R. Neale, and David M Jonas J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b09671 • Publication Date (Web): 10 Dec 2018 Downloaded from http://pubs.acs.org on December 22, 2018

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The Journal of Physical Chemistry

Carrier Dynamics and Interactions for Bulk-like Photoexcitation of Colloidal Indium Arsenide Quantum Dots Austin P. Spencer,†,‡ William K. Peters,†,¶ Nathan R. Neale,§ and David M. Jonas∗,† †Department of Chemistry, University of Colorado, Boulder, CO 80309-0215, USA ‡Current address: Department of Chemistry, Northwestern University, Evanston, IL 60208, USA ¶Current address: Los Alamos National Laboratory, Los Alamos, NM 87545, USA §Chemistry and Nanoscience Center, National Renewable Energy Laboratory, 15013 Denver West Parkway, Golden, CO 80401, USA E-mail: [email protected]

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Abstract The remarkable photonic and photochemical properties of colloidal quantum dots (QD) depend critically on the dynamics of carrier interactions and relaxation. Despite their importance, a quantitative experimental evaluation of these processes has proven elusive due to the inherent challenge of exactly separating single-exciton and multi-exciton dynamics, whose spectroscopic signatures overlap in time, spectrum, and excitation fluence. Here, we measure pump fluence–dependent absolute pump–probe transients of indium arsenide QDs, refreshing the sample using beam scanning to limit repetitive excitation. Focusing on the low fluence limit near the onset of bi-exciton formation, excitation conditions were precisely controlled and characterized by averaging Poisson-distributed excitation statistics over all three spatial dimensions of the pump and probe beam spatial profiles to determine the average excitation probability. A saturation model is developed to uniquely decompose the pump–probe signal into single-exciton and bi-exciton signals. This method harnesses the distinct pump-fluence scaling of absolute pump–probe signals from singly- and doubly-excited QDs without any assumptions regarding the relative timescales or amplitudes of single-exciton and bi-exciton signals. Probing in the bulk-like region of the QD absorption spectrum, the signal from bi-excitons is found to be 1.8 times the signal from single excitons at T = 0, consistent with the conventionally assumed factor of 2 within the 95% confidence intervals. The bi-exciton signal contains the same hot carrier relaxation dynamics as that from single excitons, but signal from a second exciton additionally exhibits a 26 ps exponential decay attributed to Auger recombination.

Introduction Colloidal quantum dots (QDs) have generated significant interest in recent decades since their uniquely tunable electronic properties make QDs appealing for such applications as solar energy harvesting (in both photovoltaic and photochemical cells), 1,2 fluorescent tagging of bio-molecules, 3,4 and illumination (for example light-emitting diodes, laser diodes, and 2


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fluorescent phosphors). 5,6 Under high current or intense illumination, quantum dots have a charge carrier loss process known as Auger recombination (AR) wherein an electron and hole recombine, transferring their excess energy to a conduction-band electron or valenceband hole. AR is the inverse of impact ionization, which is one proposed mechanism for carrier multiplication (CM) or multiple exciton generation (MEG) in an isolated dot. The strong increase in impact ionization yields with excess energy 7,8 suggests that AR might also depend on excess energy, but this has not been investigated. In this context, InAs is an intriguing material because the bulk effective masses around the direct bandgap at the Γ point 9–11 indicate that excess photoexcitation energy is partitioned mainly (94%) to the electron, with the remainder left to the hole. More generally, AR is an important and ubiquitous process in QDs since it is the dominant relaxation pathway in multiply-excited QDs and therefore fundamentally influences their photochemical, photovoltaic, and photonic properties. Despite the significant link between AR and QD performance in proposed applications, many aspects of the process remain uncharacterized. One reason for this gap in understanding is that AR has proved challenging to study in a quantitative way. Of principle concern is careful control over excitation conditions to ensure an accurate accounting of the number of excitons generated per laser pulse in each QD. Unlike molecules, QDs often have highly degenerate electronic transitions with little change between ground and excited state absorption cross sections, and so are susceptible to multiple excitation. An additional question relating to AR is whether or not pump–probe signal strength is linear with the number of excitations. Experiments typically assume that a doubly-excited QD yields twice the signal of a singly-excited QD, the validity of which is central to conclusions regarding the population of multiply-excited QDs in studies of CM. For example, pump–probe experiments often take advantage of the large difference in timescale between AR (tens of picoseconds) and single exciton recombination (hundreds of picoseconds to microseconds) to quantify the fraction of QDs that initially contained multiple excitons (after pump excitation) by comparing signal

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amplitude before and after AR. Unfortunately, when cooling processes occur on timescales similar to or slower than AR, this method is not applicable to hot carrier excitation, where Poisson statistics are most likely to hold. These experimental complications motivated the thorough evaluation of carrier relaxation and interactions in InAs QDs under well-controlled conditions using degenerate pump–probe transient absorption spectroscopy presented here. Central to this goal is characterization of single- and bi-exciton relaxation dynamics as a function of excitation probabilities that extend into the low excitation probability regime (from 125% down to 9%) using sample exchange to minimize repetitive excitation. Excitation probability, which is the probability of excitation by the pump pulse averaged over the 3D pump and probe spatial profiles, determines the proportion of QDs with zero, one, two, etc. excitations contributing to the pump–probe signal and its control is important when attempting to isolate single-, bi-, and multi-exciton dynamics from one another. It is also necessary to exchange the sample so that each pump–probe pulse sequence starts with an equilibrated sample. 12,13 Reported InAs QD single exciton lifetimes are on the order of a few nanoseconds, 14–16 so the prime concern in experiments at 10 kHz laser repetition rate is the steady-state accumulation 12 of low-yield photoproducts with long lifetimes (e.g., seconds from ”off states” 17–20 or QD photocharging 21 ). The accumulation of ”off”/photocharged QDs can be quantified by an average resampling rate. 12,13 The average resampling rate 13 (k resampling ) is the rate at which chromophores at a point in the sample are sampled by the pump during the course of the experiment. The average resampling rate is equal to the laser repetition rate (104 s−1 , here) when no sample exchange method is used (i.e. stationary sample and stationary beams), but has been reduced to 8.3 s−1 in this work through beam scanning. 13 To predict the average number of excitations per QD for a given pump pulse fluence (photon flux time-integrated over a pulse) without knowledge of the frequency dependence and time dependence of the absorption cross sections for all optically-allowed excited state absorption transitions within the laser bandwidth, assumptions must be made regarding the

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absorptive properties of the sample. If a chromophore’s probability of absorbing a photon is unaffected by its prior history of excitation such that excitation events are independent and uncorrelated, then the number of photons absorbed by an ensemble of chromophores during a period of time will obey Poisson statistics. For this to be the case, the chromophore’s absorption cross section must not be affected by excitation (i.e. the ground state and all excited states must have the same absorption cross section within the pump pulse bandwidth plus excited state stimulated emission must lie outside the laser bandwidth). Although this rarely holds for molecular systems, it is most likely to be approximately true for QDs at low excitation density in the bulk-like portions of their spectrum. 22 Assuming Poisson absorption statistics, the average number of photons absorbed per QD from a single pump pulse at each point in the sample is given by Neh = σa Φ where σa is the absorption cross section of the QD and Φ is the pulse photon fluence (photons per unit area) at that point. To reflect the experimental conditions, the frequency dependence of σa and the frequency and spatial dependence of Φ must be included and averaged over. The spatial average depends on the spatial profile of both pump and probe, resulting in the spatially-averaged excitation probability hNeh i. A handful of studies on carrier dynamics in colloidal InAs QDs have been reported. Pijpers et al. 23 investigated multi-exciton dynamics in InAs/CdSe/ZnSe core/shell/shell QDs (4.4 nm diameter core) by transient absorption spectroscopy. When pumping slightly to the blue of the “1P” transition and probing the “1S” transition, a sub-picosecond carrier cooling time and a 30 ps biexciton decay time were reported. Schaller et al. 15 performed transient absorption spectroscopy on InAs “core only” (4.3 nm diameter) and InAs/CdSe core/shell (3.9 nm diameter cores) QDs, pumping on the red side of the second exciton peak (“1P”) and probing the “1S” transition. They reported a carrier cooling time of 0.66 ps (∼0.5 ps/eV) for InAs/CdSe QDs, a biexciton lifetime of 8.3 ps for InAs “core only” QDs, and a single exciton lifetime of 192 ps for InAs “core only” QDs. Based on transient absorption measurements, pumping at 1.55 eV and probing the first exciton peak, Ben-Lulu et al. 16 report 53 ps and

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28 ps biexciton lifetimes and 3 ns and 5.2 ns single exciton lifetimes for InAs/CdSe/ZnSe core/shell/shell QDs with first exciton peaks at 0.95 eV (5.9 nm diameter cores) and 1.1 eV (4.9 nm diameter cores), respectively. Lastly, Pijpers et al. 24 report an electron relaxation time of ∼0.8 ps based on the rise of the transient absorption signal when pumping the “1P” transition and probing the “1S” transition of 4.4 nm diameter InAs QDs with a band gap of 1.1 eV. From the instrument response–limited rise time of the “1P”-pump–THzprobe signal, Pijpers et al. give an upper bound of 150 fs (or 1 − 2 eV/ps) for the hole cooling time. For the same THz measurements, biexciton time constants of 9.5 ps and 24 ps were reported in 4.0 nm diameter InAs QDs and 4.9 nm diameter InAs/CdSe/ZnSe core/shell/shell QDs, respectively. As a whole, these studies suggest a ∼0.8 ps time constant for 1Pe → 1Se electron cooling and a 10 to 30 ps time constant for Auger recombination in ligand-passivated InAs QDs with a band gap of approximately 1 eV. In this work, we resolve the pump-fluence dependence and temporal dynamics of hot carriers in the absolute pump–probe signal of InAs QDs by degenerately pumping and probing in the bulk-like spectral region above the 1Pe transition. We develop a saturation model to separate singleexciton and bi-exciton signals based on their unique pump-fluence scaling, for the first time determining the relative amplitude of single- and bi-exciton signals without assumptions regarding their relative timescales.

Experiment Sample Preparation and Characterization The synthesis of colloidal InAs QDs capped with TOP (trioctylphosphine) and/or TOPO (trioctylphosphine oxide) ligands is described in the supporting information. This synthesis is based on refs. 25, 26, and 14, but differs from those procedures in that slightly higher initial precursor concentration and smaller amounts of subsequent precursor injections were used to afford the larger particles prepared here. Also, similar to ref. 14, this procedure uses 6


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both TOP and TOPO ligands for surface passivation whereas other prior works exclusively use TOP in the InAs core synthesis. As has been shown recently by Mattousi and co-workers for CdSe-ZnS core-shell QDs, TOP weakly binds to and passivates surface heteroatoms and TOPO weakly binds to and passivates surface metal atoms. 27 Thus, the addition of TOPO to the conventional TOP solvent in InAs QD synthesis may help to passivate In and As surface sites, respectively. Anhydrous and oxygen-free conditions (see SI) were used to minimize surface oxidation. No additional surface passivation (e.g., CdSe shell) steps were taken. Characterization by transmission electron microscopy yielded a diameter distribution of 6.2 ± 0.7 nm (see SI). The linear absorption spectrum is shown in Figure 1 along with the laser pulse spectrum. The mean QD diameter is 6.2 nm based on comparison of the central wavelength of the first exciton peak to an empirical sizing curve. 14 The measured absorbance was converted to absorption cross section (σa ) using the empirical scaling of absorption cross section at the first exciton peak with QD radius reported by Yu et al. 14 The two lowest energy excitonic transitions, 1Se 1Sh at λ = 1199 nm and 1Pe 1Ph at λ ≈ 900 nm, are labeled 1S and 1P, respectively. The laser pulse spectrum, centered at λ = 793 nm and with a FWHM bandwidth of ∆λ = 21 nm, overlaps with the far-blue wing of the 1P transition. The photon energy used for experiments is below the threshold for CM.

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4 3 2 1

pump, probe

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

absorption cross section (10−15 cm2 )


1Pe 1Ph ↓ 1Se 1Sh ↓

0 800 1000 1200 1400 1600 1800 λ (nm)

Figure 1: Visible–near-infrared absorption spectrum of colloidal InAs quantum dots (blue). The first (1Se 1Sh ) and second (1Pe 1Ph ) exciton peaks are centered at λ = 1199 nm and λ ≈ 900 nm respectively. The laser spectrum (red; arbitrarily scaled) is centered at λpu,pr = 793 nm and has a full-width at half-maximum bandwidth of ∆λ = 21 nm. The small region of negative absorbance near λ = 1700 nm is the result of imperfect background subtraction of solvent (toluene) absorption features.

Pump–Probe Spectroscopy The output of a Ti:Sapphire regenerative amplifier (Coherent RegA; λ = 793 nm, 10 kHz repetition rate, 85 fs pulse duration) is split into pump and probe beams. An optical chopper (3501, New Focus) that is synchronized (i.e. phase-locked) to the 4th subharmonic of the laser repetition rate amplitude modulates the pump beam at 2.5 kHz (two pulses transmitted and two pulses blocked in each cycle) and is phased such that pulse clipping is minimized. Pump and probe beams then enter an interferometer that controls the time delay (T ) of the probe pulses relative to the pump pulses. From the interferometer, parallel (but non-collinear) beams propagate into a beam scanning apparatus 13 that enables average resampling rates of 8.3 s−1 for the measurements reported here. The all-reflective optics beam scanning apparatus uses a fast steering mirror (FSM) to move the pump–probe focal spot in a repeating spiral pattern with respect to the stationary sample such that the laser photon flux (and consequently the time-averaged excitation) is spread over a ∼6.3 mm2 area of the sample 8


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cell, reducing repetitive excitation. Pump and probe beams are focused to a common point with a beam focal spot intensity FWHM (d1/2 ) of 48 µm (see SI). The electrical signal from the probe photodiode is filtered and amplified by a gated integrator (SR250, Stanford Research Systems) which time-integrates the single pulse waveform from the photodiode during a short time window (on the order of the response time of the detector) and amplifies the voltage. The output is then routed to a lock-in amplifier (SR830, Stanford Research Systems) that is referenced to the optical chopper. The lock-in amplifier isolates and amplifies components of its input waveform that oscillate at the same frequency as, and in phase with, the optical chopper. The resulting pump–probe signal is read digitally from the lock-in amplifier by a PC executing a LabVIEW program that controls data collection. The signal is averaged 3–10 times at each time delay in a scan and 3–25 scans are collected and averaged. The beam scanning apparatus used to minimize repetitive excitation artifacts is fully described in ref. 13. A discussion of how to minimize such artifacts in various photophysical regimes is given in ref. 12. Because the excited state lifetime is five orders of magnitude shorter than the laser repetition period, a 23% shot-to-shot pulse overlap is acceptable and the pattern is designed to minimize the resampling rate in order to prevent steady-state accumulation of low-yield long-lived photoproducts often attributed to photocharging. All influence of shot-to-shot overlap is neutralized by phased lock-in detection at klaser /4. 13 As in ref. 13, the scan pattern fills a circular annulus with ∼3 mm outside diameter and ∼1 mm inside diameter over a 0.5 s period, yielding an average resampling rate of 8.3 s−1 . Saturation experiments at 1 ns pump–probe delay were performed with two different resampling rates in order to determine whether this resampling rate was sufficiently low for the InAs QDs studied here (see SI). These measurements put an upper bound of 2 ms on the quantum yield– lifetime product (φLLS × τLLS ) for photocharging of this InAs QD sample; with a 8.3 s−1 resampling rate, Eq. S1 of the SI predicts that accumulated steady-state photocharging would be effectively suppressed to below a few percent even at excitation probabilities of

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200%. Pump–probe transients from T = −1 ps to 1 ns were collected with a 1 nJ probe pulse energy (Epr ) and at pump pulse energies (Epu ) ranging from 0.45 nJ to 6.23 nJ (see Table 1). The pump–probe signal was confirmed to be linear with respect to sample optical density (to within 10% relative to the signal maximum) at both short (T = 0) and long (T = 1 ns) time delays for optical densities of 0.016 and 0.10, although at an optical density of 0.37 this was no longer the case. Scaling of pump–probe signal with probe pulse energy was also tested. At a probe pulse energy of 1 nJ, the signal was found to be 99 ± 15% of the linear extrapolation from low pulse energy. 28 The probe-weighted spatially-averaged excitation probability induced by the pump pulse was calculated for each experimental pump pulse energy. For simplicity, pump and probe beams were approximated such that they (a) overlap perfectly through the entire sample cell, (b) have identical circular Gaussian intensity profiles, (c) have identical spectra, (d) are well-collimated over the length of the sample cell (Rayleigh range zR = 6.7 mm), (e) are attenuated exponentially as a function of depth into the sample (i.e. pump and probe pulse absorption each individually obey the Beer-Lambert law so that their pulse propagation can be described with linear optics even though the small effect of the pump on probe transmission is a nonlinear optical effect), and (f) have a frequency bandwidth that is small compared to both their central frequency and the bandwidth of overlapping features in the sample absorption spectrum such that the pulse bandwidth can be neglected (variation in optical density within the pulse FWHM bandwidth is less than 5%). Under these approximations, the photon fluence of pump and probe beams are described by Gaussian beam spatial profiles (chapter 17.1, equation 6 of Siegman), 29 given by 2r2 2 exp − 2 exp (−αz) h(r, z) = πw2 w

(1)

where r is the radial polar coordinate perpendicular to the axis along which the beam

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propagates (z), w is the beam waist or focused spot size [h(r = w, z = 0)/h(r = 0, z = 0) = 1/e2 ], α = [OD ln(10)]/L is the intensity attenuation coefficient at the center of the pulse spectrum, OD is the sample optical density, and L is the sample path length. The beam √ spot size is related to the FWHM diameter (d1/2 ) of beam intensity through w = d1/2 / 2 ln 2 so that d1/2 = 48 µm corresponds to w = 41 µm. (Note that there is a change in notation from ref. 13; d1/2 here is equal to wFWHM of ref. 13. This change was made to use waist w 0 consistently with ref. 29 here.) The beam spatial profile at the sample cell entrance (z = 0) R∞ has unit area: 0 h(r, z = 0)2πrdr = 1 where the factor of 2π arises from integration over the angle θ, on which the beam spatial profile does not depend. Multiplying eq. 1 by the 0 = U0 /(~ω) where U0 is the incident pump pulse number of incident pump pulse photons [Npu

energy] yields the pump pulse photon fluence

0 Φpu (r, z) = Npu hpu (r, z) =

U0 hpu (r, z) ~ω

(2)

where variables with a subscript “pu” relate to the pump pulse. Subsequent multiplication of the pump pulse photon fluence in eq. 2 by the absorption cross section (σa ) of the QD sample at the central frequency of the pump pulse gives the excitation probability as a function of position within the sample Neh (r, z) = Φpu (r, z) σa .

(3)

The number of absorbed photons per QD is assumed to obey Poisson statistics although this cannot be rigorously true since it predicts that, in the absence of fast relaxation during the pulse, there would be no pump–probe signal during pulse overlap, in contradiction to experimental observation. On the other hand, there is no measurable deviation (less than 1%) from the linear relationship between pump absorption and pump pulse energy (see SI) predicted by Poisson statistics, although Poisson statistics are not required for such linear absorption versus pulse energy. (For example, a harmonic oscillator has no nonlinear response and the same net linear absorption for all pulse energies, but has different level

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population statistics. Non-interacting carriers have the evenly spaced energy levels of a harmonic oscillator.) The probability of a QD having n excitations,

Pn (r, z) =

[Neh (r, z)]n exp [−Neh (r, z)] , n!

(4)

is then calculated using the Poisson distribution with Neh as the mean. The Poisson probability coefficients for n = 0, 1, 2, . . . in eq. 4 are spatially averaged over the probe beam spatial profile R∞ hPn ir,z =

2πrdr

RL

dz Pn (r, z) hpr (r, z)

0

0

R∞

2πrdr

RL

0

(5) dz hpr (r, z)

0

by numerical integration. The subscript “pr” on the right-hand side in eq. 5 indicates that the spatial average is weighted by the probe spatial profile. Equations 1 and 5 show that in experiments with wpr wpu and αL 1, hPn ir,z approaches Pn (0, 0) over spatial regions with appreciable probe fluence. This approach has been used 30 for static and flowing quantum dot samples. Note that even if the underlying probability of excitation at each point in the sample has a Poisson distribution, the probe-weighted spatially-averaged Poisson coefficients hPn ir,z do not obey a Poisson distribution, although they approach the underlying distribution as U0 goes to zero. Finally, the probe-weighted spatially-averaged excitation probability hNeh ir,z =

∞ X

n hPn ir,z

(6)

n=1

is calculated. The probe-weighted spatially-averaged excitation probability is linearly proportional to the pump pulse energy: hNeh ir,z = (0.201 nJ−1 )U0 for the conditions of these experiments, in which bulk-like excitation prevents band-filling. Table 1 summarizes the excitation conditions predicted by the above-described spatiallyaveraged Poisson excitation probability model for the pump pulse energies used in this work. hNeh ir,z ranges from 0.09 to 1.25 and follows a linear relationship relative to the

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excitation probability in the center of the pump beam at the sample entrance, Neh (0, 0). The fraction of pump-excited QDs with two excitations (i.e., initially-excited biexcitons), given by hP2 ir,z / (1 − hP0 ir,z ), is significantly less than the fraction of singly-excited QDs for all pump pulse energies and much less for most, indicating that these experiments probe the low–photon-flux threshold region for biexciton generation through double excitation. P nhPn ir,z / m mhPm ir,z gives the fraction of the pump–probe signal arising from QDs with n initial excitations under the assumption that signal scales linearly with the number of excitons. While single-excitons dominate the overall pump–probe signal at the lowest pulse energies, multi-excitons generate the majority of the signal at the two highest pulse energies. Table 1: Experimental incident pump pulse energies U0 , maximum excitation probability Neh (0, 0), and probe-weighted spatially-averaged excitation probabilities hNeh ir,z (eq. 6). hPn ir,z (eq. 5) is the spatially-averaged probability of exciting a QD n times based on Poisson absorption statistics at each point in the sample. hPn iP r,z /(1 − hP0 ir,z ) is the fraction of excited QDs that have n excitons and nhPn ir,z / m mhPm ir,z is the fraction of signal arising from QDs with n total excitons on the assumption that the signal is linearly proportional to the number of excitons. Absorption cross section, σa (λ = 793 nm) = 2.979 × 10−15 cm2 ; beam spot size, w = 41 µm; optical density, OD(λ = 793 nm) = 0.10.

U0 (nJ)

Neh (0, 0) hNeh ir,z

hP0 ir,z

hP1 ir,z

hP2 ir,z

hP3 ir,z

0.45 1.00 1.50 1.99 3.74 6.23

0.20 0.45 0.67 0.89 1.67 2.79

0.92 0.82 0.75 0.69 0.52 0.37

0.080 0.15 0.20 0.24 0.29 0.28

4.8 × 10−3 0.020 0.039 0.059 0.13 0.18

2.1 × 10−4 2.0 × 10−3 5.7 × 10−3 0.012 0.044 0.097

U0 (nJ)

hP1 ir,z 1−hP0 ir,z



P

0.45 1.00 1.50 1.99 3.74 6.23

94% 87% 82% 77% 61% 45%

5.6% 11% 16% 19% 26% 29%

0.25% 1.1% 2.3% 3.7% 9.1% 15%

89% 77% 68% 60% 39% 23%

0.090 0.20 0.30 0.40 0.75 1.25

13

hP1 ir,z nhPn ir,z

n


2hP2 ir,z P n nhPn ir,z

3hP3 ir,z P n nhPn ir,z

11% 20% 26% 30% 34% 29%

0.7% 3.0% 5.7% 8.6% 18% 23%


Results and Discussion Absolute Pump–Probe Transients Absolute pump–probe transients over the full 1 ns range of time delays are shown in Figure 2. The absolute pump–probe signal is the change in the number of transmitted probe photons (∆N ) caused by prior pump excitation (compared to without pump excitation) divided by the total number of incident probe photons (N0 = 4.0 × 109 photons). No ad hoc scaling or “tail matching” is employed. Pump–probe transients at a range of excitation probabilities were collected in the following order: hNeh ir,z = 75%, 125%, 30%, 20%, 9.0%, and 40%. To correct for small accumulated errors in delay stage position due to occasional mis-steps (a second harmonic autocorrelation recorded immediately after the experiment showed time zero changes by +20 fs compared to the time zero immediately before the experiment), the pump–probe delay axes were adjusted such that the rise of the signal coincides between transients. This involved a +10 fs shift for transients with hNeh ir,z = 40% and hNeh ir,z = 125% as well as a +20 fs shift for the hNeh ir,z = 75% transient. An increase in signal size with increasing excitation probability is visually apparent. When normalized by excitation probability, as in the bottom panel of Figure 2, the four lowest excitation transients are in good agreement. However, the 75% and 125% excitation probability transients lie slightly below the others starting at time zero (see inset in bottom panel of Figure 2), possibly indicating saturation of the pumped transition. This trend persists through T = 1 ns.

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(∆N/N0 ) × 103

1.4 1.2

1

1

0.5

0.8

0

0

5

hNeh ir,z 125% 75% 40% 30% 20% 9.0%

10 15 20

0.6 0.4 0.2 0

0

200

400 600 delay (ps)

1.2 1

[(∆N/N0 )/hNeh i] × 103

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1

0.5

0.8

0 −1

0

1

2

0.6

800

1000

hNeh ir,z 9.0% 20% 30% 40% 75% 125%

0.4 0.2 0

0

5

10 delay (ps)

15

20

Figure 2: Pump–probe transients for probe-weighted spatially-averaged excitation probabilities ranging from 9% to 125%. In the top panel, signal (∆N/N0 ) is plotted as the change in number of transmitted probe photons per incident probe photon, multiplied by a factor of 103 , with an inset highlighting the initial 20 ps. In the bottom panel, the signal is divided by the spatially-averaged excitation probability (hNeh ir,z ) with an inset highlighting the initial 2 ps.

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Saturation Model To model the power dependence of the signal, transients were fit to a two-level system saturation model. This model relates pump pulse energy (or, equivalently, excitation probability) to signal amplitude through

sat Spp (U0 ; x, y, T ) =

x(T )U0 1 + y(T )U0

(7)

sat (U0 ; x, y, T ) is the pump–probe signal, x(T ) is the slope of pump–probe signal with where Spp

pump pulse energy in the low pulse energy limit, U0 is the incident pump pulse energy, and y(T ) = 1/Usat (T ) is the inverse of the saturation pump pulse energy. When U0 = Usat (T ), the signal is half of its linear extrapolation from low energy given by

lin Spp (U0 ; x, T ) = x(T )U0 ,

(8)

which we term the linear reconstruction. Equation 8 is the first-order term of the Taylor series expansion of eq. 7 around U0 = 0. The linear reconstruction is expected to contain dynamics from QDs that absorb one pump photon (single exciton QDs) since these signal components are linear in pump excitation. Differences between the saturated signal (eq. 7) and the linear reconstruction (eq. 8) are indicative of signal components that are not linear with pump excitation and can arise from a number of sources, including nonlinear optical saturation of a pumped transition (for saturation at early time delays), Auger recombination (for saturation at late time delays), and repetitive excitation effects. The saturation model (eq. 7) was fit to the series of transients independently at each pump–probe time delay T (see SI). Examples of saturation fits to the pump–probe signal at T = 500 fs, T = 20 ps, and T = 1 ns are shown in Figure 3. As U0 and hNeh ir,z are linearly related, saturation fits in this study were calculated as a function of hNeh ir,z . Reported values of x(T ) and y(T ) are in units of (∆N/N0 )/hNeh ir,z

16


Page 17 of 38

and 1/hNeh ir,z , respectively. 1.6 T = 500 fs T = 20 ps T = 1 ns saturation fit linear reconstruction

1.4 1.2 (∆N/N0 ) × 103

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1 0.8 0.6 0.4 0.2 0

0

0.5

1 hNeh ir,z

1.5

Figure 3: Pump–probe signal, ∆N/N0 , measured as a function of probe-weighted spatiallyaveraged excitation probability, hNeh ir,z at pump–probe delays of T = 500 fs (blue circles), T = 20 ps (green squares), and T = 1 ns (red triangles) with saturation fits (solid lines) and the corresponding linear reconstructions (dashed lines). Sample path length, L = 1 mm; optical density, OD(λ = 793 nm) = 0.1; laser repetition rate, klaser = 10 kHz; beam spot size, w = 41 µm; average resampling rate, k resampling = 8.3 s−1 . The probe-weighted spatially-averaged probability of a QD being excited at least once initially by the pump pulse is given by 1 − hP0 ir,z . Since Auger recombination is significantly faster than single exciton recombination 15 through radiation or internal conversion, at pump– probe delays much longer than the Auger recombination lifetime but much shorter than the single exciton recombination lifetime, all initially excited QDs have exactly one exciton regardless of the initial number of excitations. In this regime of pump–probe delays, the pump pulse energy dependence of the signal is expected to go as 1 − hP0 ir,z (see supporting info). Even at delays for which significant single exciton recombination may have taken place, as long as the timescale of AR is much shorter than both T and the timescale of single exciton recombination, the pump–probe signal is expected to be proportional to 1 − hP0 ir,z . 17



To test this expectation, the pump–probe signal at T = 1 ns is compared to 1 − hP0 ir,z in Figure 4. In the low excitation probability limit (below the lowest excitation probability used here), hNeh i becomes equal to 1 − hP0 ir,z , resulting in a linear relationship between Spp (T ) and hNeh i. When 1 − hP0 ir,z is multiplied by x(1 ns) such that the linear components of 1 − hP0 ir,z and Spp (1 ns) are equal (green dashed line), it is clear that x(1 ns) · (1 − hP0 ir,z ) falls below Spp (1 ns) for hNeh ir,z > 1. When 1 − hP0 ir,z is instead scaled by a multiplicative factor a (red solid line) that minimizes the least square residual relative to Spp (1 ns), we observe good qualitative agreement in pump pulse energy scaling. While x(1 ns) < a, this deviation is not statistically significant since a falls within the 95% confidence interval of x(1 ns). This agreement corroborates the excitation statistics calculated using Poisson absorption statistics and spatial averaging.

18


Page 18 of 38

Page 19 of 38

0.08 Spp (1 ns) a · (1 − hP0 ir,z ) x(1 ns) · (1 − hP0 ir,z )

0.06 (∆N/N0 ) × 103

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.04

0.02

0

0

0.5

1

1.5

2

hNeh ir,z Figure 4: Pump–probe signal, ∆N/N0 , measured as a function of probe-weighted spatiallyaveraged excitation probability, hNeh ir,z at a pump–probe delay of T = 1 ns (blue circles) compared to the fraction of QDs initially excited by the pump pulse, 1 − hP0 ir,z , multiplied by the best fit parameter a = 7.7 ± 0.3 · 10−5 (solid red line) or by x(1 ns) = 6.8 ± 1.7 · 10−5 (dashed green line) from a saturation fit to the shown pump–probe signal data. The chosen value of a minimizes the residuals between a · (1 − hP0 ir,z ) and Spp (1 ns) in a linear least squares sense. Sample path length, L = 1 mm; optical density, OD(λ = 793 nm) = 0.1; laser repetition rate, klaser = 10 kHz; beam spot size, w = 41 µm; average resampling rate, k resampling = 8.3 s−1 .

19



Using the fit parameters x(T ) and y(T ) at each time delay T , the linear signal component (eq. 8) can be calculated. This reconstruction agrees well at early pump–probe delays with the hNeh ir,z = 0.20 transient, as shown in top panel of Figure 5, and the hNeh ir,z = 0.090 transient. At late pump–probe delays (T > 500 ps), however, the linear component lies 6% and 12% higher than the saturation fit for hNeh ir,z = 0.090 and hNeh ir,z = 0.20, respectively. This comparison suggests that the transients with the lowest probe-weighted spatially-averaged excitation probabilities are dominated by single exciton dynamics but contain small—yet measurable—biexciton contributions that decay through AR. This conclusion agrees quantitatively with the predicted late-T signal saturation (relative to the linear component) due P to AR, which is given by (1 − P0 )/ nPn . In contrast, the linear component overestimates the signal size of the hNeh ir,z = 1.25 transient by ∼15% at the earliest delays, as shown in the bottom panel of Figure 5, indicating significant saturation of pump excitation at the highest pump pulse energy used. This initial saturation implies the initial excitation number distribution is not Poisson at the highest pump pulse energy.

20


Page 20 of 38

Page 21 of 38

(∆N/N0 ) × 103

0.25

data saturation fit linear component

0.2 0.15 0.1 0.05 0

0

5

10 delay (ps)

1.5 (∆N/N0 ) × 103

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


15

20

data saturation fit linear component

1

0.5

0

0

5

10 delay (ps)

15

20

Figure 5: Pump–probe transients (blue circles) with hNeh ir,z = 20% (top panel) and hNeh ir,z = 125% (bottom panel) and their respective linear components (dashed green line) from saturation model fits (solid red line) at each time delay. Sample path length, L = 1 mm; optical density, OD(λ = 793 nm) = 0.1; laser repetition rate, klaser = 10 kHz; beam spot size, w = 41 µm; average resampling rate, k resampling = 8.3 s−1 .

21



Page 22 of 38

Reconstructed Signals for Initially Singly- and Doubly-Excited Particles To separate pump-probe signals for QDs that are initially singly-excited and doubly-excited by the pump pulse, a procedure relying on the distinct scaling of initially excited multiexcitonic signal components with pump pulse energy was developed. 28 Saturation fits to the pump–probe transients at each pump–probe delay are taken as a complete description of the time and pump pulse energy dependence of the experimental pump–probe signal. The Taylor series expansion of the saturation model is given by

sat Spp (T ) =

∞ X

x(T ) [−y(T )]n−1 U0n (9)

n=1

= x(T ) U0 − x(T )y(T ) U02 + . . . where x(T ) and y(T ) are the saturation model parameters from eq. 7 and U0 is the incident pump pulse energy. The probe-weighted spatially-averaged probability of a QD receiving one (hP1 ir,z ) or two (hP2 ir,z ) pump-induced excitations can be written as power series hP1 ir,z = c1 U0 − c2 U02 + . . .

(10)

1 hP2 ir,z = c2 U02 + . . . 2

(11)

and

where cn are constants derived from a polynomial fit to numerically calculated values of hP1 ir,z . For the saturation model (eq. 7), the value of c2 is positive such that a QD’s second excitation eliminates a single exciton in eq. 10 and generates a biexciton in eq. 11.

22


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Equations 10 and 11 are constructed such that by eq. 6, hNeh ir,z = hP1 ir,z + 2hP2 ir,z + . . . (12) = c1 U0 . We define = c1 U0 hP1 ilinear r,z 1 hP2 iquad = c2 U02 r,z 2

(13)

such that hP1 ir,z = hP1 ilinear − 2hP2 iquad r,z r,z + . . . hP2 ir,z =

hP2 iquad r,z

(14)

+ ...

A theoretical form for the pump–probe signal is constructed from eq. 14, yielding theory Spp = b {hP1 ir,z E(T ) + hP2 ir,z B(T ) + . . .} quad = b hP1 ilinear r,z E(T ) + hP2 ir,z [B(T ) − 2E(T )] + . . .

(15)

where E(T ) is the pump–probe signal from one initially singly-excited QD (single exciton) and is peak-normalized, B(T ) is the pump–probe signal from one initially doubly-excited QD (biexciton), and b is a constant specific to the experimental setup and detection sensitivity. The height of B(T ) is not constrained, thus allowing the common assumption that the initial pump–probe signal is strictly linear with the total number of excitations and is insensitive ?

to their distribution [i.e. B(0) = 2E(0)] to be tested. Substituting eq. 13 into eq. 15 yields

theory Spp

1 2 = b c1 U0 E(T ) + c2 U0 [B(T ) − 2E(T )] + . . . . 2

23


(16)


Page 24 of 38

Equating the terms in eqs. 9 and 16 that are linear in U0 and solving for E(T ) gives 1 1 E(T ) = x(T ) . b c1

(17)

Doing the same for the quadratic terms and solving for B(T ) gives 2 1 1 B(T ) = x(T ) − x(T )y(T ) . b c1 c2

(18)

lin Comparing eqs. 8 and 17, it is noted that Spp (T ) and E(T ) differ only by a constant multi-

plicative factor and, therefore, have the same pump–probe delay dependence. E(T ) and B(T ) in Figure 6 are calculated according to eqs. 17 and 18 using experimentally measured x(T ) and y(T ) along with coefficients cn derived from a smooth 9th -order polynomial fit to the calculated hP1 ir,z for an underlying Poisson distribution. For the 9th order polynomial, the lowest order coefficients are numerically converged, not significantly affected by increases in polynomial order, and obey relationships expected for the polynomial coefficients of hP2 ir,z , etc. Both E(T ) and B(T ) appear to contain many of the same decay timescales in similar proportions, as expected given that initially excited biexcitons also undergo single exciton processes such as cooling and recombination prior to, during, and after AR. At long pump–probe delays (T > 300 ps), E(T ) and B(T ) are equal within their uncertainties, implying that long after AR, all initially doubly-excited QDs have relaxed to single exciton states. In contrast, at short pump–probe delays (T < 50 ps), B(T ) lies above E(T ), reflecting the additional signal contributed by a second exciton. Since the amplitude of B(T ) is not constrained, its peak height with respect to E(T ) is indicative of the relative signal contribution of a biexciton compared to a single exciton. This allows a test of the common assumption that a biexciton contributes exactly twice the signal of a single exciton. Figure 6 suggests that a biexciton initially yields 1.8±0.4 times as much signal as a single exciton when probed in the bulk-like region of the QD absorption spectrum. The timescale on which B(T ) converges to E(T ) is indicative of AR and potentially other processes requiring 24


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two excitons. Extraction of this timescale is discussed below. 2

B(T ) E(T )

1.5 1 0.5 0 0

5

10 delay (ps)

15

2

20

B(T ) E(T )

1.5 1 0.5 0 0

200

400 600 delay (ps)

800

1000

Figure 6: Reconstructed signals for initially singly-excited, E(T ), (blue line) and doublyexcited, B(T ), (green line) QDs with 95% confidence intervals (dotted lines). Sample path length, L = 1 mm; optical density, OD(λ = 793 nm) = 0.1; laser repetition rate, klaser = 10 kHz; beam spot size, w = 41 µm; average resampling rate, k resampling = 8.3 s−1 .

Single Exciton Dynamics The signal for initially singly-excited QDs, E(T ), is well described by the sum of a 4exponential decay and an exponentially-damped cosine given by 4 X

T T + D cos [2πνD T ] exp − Spp (T ) = An exp − τn τD n=1

25


(19)


with the parameters listed in Table 2. Multi-exponential fits to the raw data in Figure 2 recover similar time constants (differing by up to 25%); since the sum of a few exponentials can effectively fit a non-exponential decay or a continuous distribution of exponential decays, the time constants may not all be individually physically significant. The decaying cosine barely survives pulse overlap (when coherent responses, such as from the solvent, may contribute), but could also have contributions from optical phonons 31–33 if phonon dephasing is distorted by hot-carrier relaxation. The 4-exponential fit to E(T ) is illustrated in Figure 7. The fastest time constant in the fit (τ = 0.77 ps) is consistent with prior reports of the 1P → 1S electron cooling time. Schaller et al. 15 report a carrier cooling time of 0.66 ps in 3.9 nm diameter InAs/CdSe core/shell QDs based on the rise of “1S” signal when pumping the “1P” state. Similarly, Pijpers et al. 24 report a 0.8 ps “1P to 1S” electron relaxation time in 4.4 nm diameter InAs QDs. The authors attribute this timescale to electron–hole coupling, proposing that electrons, as a result of their comparatively wide electronic level spacing, primarily cool through an Auger process in which a hot electron transfers energy to a hole that subsequently cools through phonon emission. 24 The longest time constant (τ = 1.0 ns) is attributed to single exciton recombination. Reports of single exciton lifetimes vary widely and are highly dependent on the QD surface chemistry, their exposure to oxidation, and the details of their synthesis and handling. Single exciton lifetimes from 192 ps for 4.3 nm diameter InAs “core only” QDs 15 to 5.2 ns for 4.9 nm diameter InAs/CdSe/ZnSe core/shell/shell QDs 16 have been reported. Radiative lifetimes calculated from the Strickler-Berg relations are longer than both reported lifetimes, 14 suggesting additional non-radiative decay. If this non-radiative decay arises mainly from incomplete surface passivation, the radiative lifetime size scaling in ref. 14 suggests that the TOP/TOPO ligand capped 6.2 nm diameter InAs QDs studied here are more effectively passivated than the TOP ligand-capped 4.3 nm diameter InAs QDs studied in ref. 15, but not as fully passivated as the 4.9 nm diameter InAs/CdSe/ZnSe core/shell/shell QDs in ref. 16. The intermediate time constants (τ = 5.1 ps and τ = 75 ps) in the single exciton (low

26


Page 26 of 38

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pump pulse energy) signal are not reported in prior studies, which—unlike this work—probe the 1Se 1Sh transition. These time constants may indicate that probing in the bulk-like region of the QD absorption spectrum is sensitive to relaxation, carrier trapping, or trap recombination processes that are not probed at the 1Se 1Sh transition. Carrier trapping processes can be highly heterogeneous, and even widely separated time constants might be effectively fitting an inhomogeneous distribution of quantum dots that each have a single exponential decay from trapping (see ref. 34). This heterogeneous interpretation suggests partial surface passivation with QD-to-QD variation in the number or type of trap recombination sites. Table 2: 4-exponential with a damped cosine fit (eq. 19) to the signal for initially singly-excited QDs E(T ). Damped cosine parameters: D = 0.061 ± 0.027, νD = 6.6 ± 0.8 ps−1 , and τD = 0.11 ± 0.09 ps.

A

τ (ps)

0.50 ± 0.03 0.19 ± 0.03 0.12 ± 0.01 0.133 ± 0.005

0.77 ± 0.08 5.1 ± 1.2 75 ± 15 1061 ± 1

27



1 0.8 E

0.6 0.4 0.2 0 −0.5

0

0.5

1

1.5

2

delay (ps) 1 0.8 E

0.6 0.4 0.2 0

0

5

10 delay (ps)

15

20

1 0.8 0.6 E

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 38

0.4 0.2 0

0

200

400 600 delay (ps)

800

1000

Figure 7: Reconstructed signal for initially singly-excited QDs, E(T ), (blue circles) and a 4-exponential with damped cosine fit (red line). Sample path length, L = 1 mm; optical density, OD(λ = 793 nm) = 0.1; laser repetition rate, klaser = 10 kHz; beam spot size, w = 41 µm; average resampling rate, k resampling = 8.3 s−1 .

28


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Multi-exciton Dynamics In bulk-like regions 22 of a QD’s spectrum, it is a reasonable approximation to treat the pump– probe signal size as directly proportional to the number of electron–hole pairs (excitons) in the sample. In such a case, the signal generated by a QD in a biexciton state is twice that of a QD in a single exciton state. A common method for quantifying the yield of biexcitons generated by the pump pulse is to compare the signal size at pump–probe delays that are short compared to AR, but long compared to carrier cooling, to the signal size at pump–probe delays that are long compared to AR, but short compared to single exciton recombination. 35 However, this method will not work here due to the presence of single exciton relaxation pathways with time constants (τ = 5.1 ps and τ = 75 ps) that are on the order of the expected AR time constant (τAR = 10 to 30 ps). In addition, this method relies heavily on the assumption that pump–probe signal is strictly linear with the number of excitons which, although verified here for bulk-like excitation of hot carriers, may not be justified when pumping and/or probing an excitonic transition. Instead, an alternative method is proposed. Given the ability to separate the dynamics of initially doubly- and singly-excited QDs into B(T ) and E(T ), the processes unique to biexcitons can be isolated through an algebraic combination of B(T ) and E(T ). If it is assumed that

E(T ) = X(T ) (20) B(T ) = XX(T ) X(T ) + X(T ) where X(T ) contains the signal for processes involving exactly one exciton and is peak normalized and XX(T ) contains the signal for processes involving exactly two excitons, then the dynamics specific to two-exciton processes can be extracted using

XX(T ) =

B(T ) − 1. E(T )

29


(21a)


Page 30 of 38

Substituting eqs. 18 and 17 into eq. 21a yields c1 XX(T ) = 2 1 − y(T ) − 1 c2 c1 = 1 − 2 y(T ). c2

(21b)

To account for the slight and experimentally insignificant difference in amplitude between B(T ) and E(T ) at late times, the ratio B(1 ns)/E(1 ns) = 1.13 derived from the data in Figure 3 is subtracted from B(T )/E(T ) instead of 1.

XX(T ) =

B(T ) B(1 ns) − E(T ) E(1 ns)

(21c)

The result is illustrated in Figure 8 along with a single exponential fit with τ = 26 ± 5 ps. This time constant is consistent with AR and is well within the 8 ps to 53 ps range of reported biexciton lifetimes for InAs QDs. 15,16,23,24 The closest comparison in terms of QD core size is a study by Ben-Lulu et al. 16 who report a 53 ps biexciton lifetime for 5.9 nm diameter InAs/CdSe/ZnSe core/shell/shell QDs. However, their observation of a relatively long biexciton lifetime may be due to the use of core/shell structures as opposed to the as-prepared ligand-capped QDs used in the present study. The most comparable study of similar sized as-prepared ligand-capped InAs QDs is that of Schaller et al. 15 wherein a 8.3 ps biexciton lifetime was reported for 4.3 nm diameter InAs “cores”. If AR time constants scale with QD volume, 36,37 a 8.3 ps biexciton lifetime in 4.3 nm QDs would predict a 27 ps biexciton lifetime for the 6.2 nm QDs of the present study. While Schaller et al. 15 note no difference in pump–probe signal size between transients collected while the sample is “static” compared to when the sample is stirred, insufficient sample renewal can result in repetitive excitation effects such as photocharging which can complicate studies of AR. 38 Photocharged QDs can undergo trion decay, which is the nonradiative recombination of an exciton through transferring excess energy to a free charge carrier. Since trion decay occurs on timescales similar to AR, it can artificially increase the 30


Page 31 of 38

apparent biexciton yield when inadequate sample renewal leads to photocharging. 38 0.8 0.6 XX(T )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.4 0.2 0 −0.2 −0.4 0

20

40 60 delay (ps)

80

100

Figure 8: The ratio of signals for initially doubly- and singly-excited QDs, given by eq. 21c, (blue line) and a single exponential fit (red line). The ratio varies widely at long times where E(T ) and B(T ) tend to zero. Sample path length, L = 1 mm; optical density, OD(λ = 793 nm) = 0.1; laser repetition rate, klaser = 10 kHz; beam spot size, w = 41 µm; average resampling rate, k resampling = 8.3 s−1 .

Conclusions Carrier relaxation dynamics of 6.2 nm diameter colloidal InAs QDs in the weak excitation regime have been investigated using degenerate pump–probe transient absorption spectroscopy. A beam scanning apparatus was employed to reduce repetitive excitation of the sample, reducing the resampling rate from the 10 kHz laser repetition rate to 8.3 s−1 . This resampling rate is low enough to prevent signal from accumulated photoproducts such as charged QDs. Electronic relaxation following photoexcitation to the blue side of the 1Pe 1Ph state was characterized. A saturation model was fit to transients with excitation probabilities ranging from hNeh i = 9.0% to 125%, from which the linear signal was reconstructed. A comparison between the linear reconstruction and a transient with hNeh i = 20% demonstrates that this measurement is dominated by dynamics of singly excited QDs. 31



A procedure for isolating dynamics from initially singly- and doubly-excited QDs was developed. This method utilizes the Taylor series expansion of the saturation model along with Poisson-based calculations of the probe-weighted spatially-averaged excitation probability and relies on fewer assumptions than previously reported methods. 35 Comparison of reconstructed singly-excited, E(T ), and doubly-excited, B(T ), QD signals near T = 0 indicates that a biexciton yields a factor of 1.8 ± 0.4 times the signal of a single exciton when probed in the bulk-like region of the QD absorption spectrum; the conventionally assumed factor of 2 is consistent within the 95% confidence intervals for B(T ). This factor of ∼2 for probing in the bulk-like region of the spectrum does not imply that the same factor holds for bandgap probing; quantum confined excitonic transitions may behave differently than bulk-like transitions. 22 At long pump–probe delays (T ≥ 300 ps), B(T ) and E(T ) converge to the same value within their uncertainties, lending evidence to the expectation that all initially excited biexcitons recombine through a sub–100 ps process to form single excitons at late times. The reconstructed signal for initially singly-excited QDs was well described by a 4exponential plus damped cosine fit, yielding time constants of 0.77 ± 0.08 ps and 1061 ± 1 ps, consistent with electron cooling and single exciton recombination respectively, and two previously unreported time constants of 5.1 ± 1.2 ps and 75 ± 15 ps that may be the result of hot carrier cooling or trapping. These time constants might be an effective fit to a broad inhomogeneous distribution of trapping times. The reconstructed signal for initially doublyexcited QDs contains, in addition to the single exciton dynamics, a time constant of 26 ± 5 ps (for 6.2 nm diameter QDs) that is consistent with Auger recombination. The AR lifetime reported here for 6.2 nm diameter ligand-capped QDs without an inorganic shell agrees, within error, with QD volume scaling 36,37 of the 8.3 ps literature AR lifetime for 4.3 nm diameter 15 ligand-capped InAs QDs.

32


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Supporting Information Available Materials and synthesis of InAs quantum dots, TEM images, measurement of laser beam focal spot size, tests that pump pulse absorption scales linearly with pump pulse energy, estimates of the quantum yield–lifetime product parameter for photoproducts from a repetitive excitation kinetic model, tests of the saturation model, and saturation model fits x(T ) and y(T ).

Acknowledgement We thank Kyle Schnitzenbaumer for help in measuring TEM images of the sample. This material is based upon work of A.P.S., W.K.P., and D.M.J. supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences, under Award Number DE-FG02-07ER15912. This work was authored in part by Alliance for Sustainable Energy, LLC, the manager and operator of the National Renewable Energy Laboratory for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. Support for InAs QD synthesis was provided by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, Chemical Sciences, Geosciences, and Biosciences Division.

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